DOI: https://doi.org/10.15446/recolma.v49n2.60445

Free subgroups of the parametrized modular group

Subgrupos libres del grupo modular parametrizado

Christian Pommerenke¹, Margarita Toro^0,2

⁰ The second author was partially supported by the proyect "Matemáticas y computación", Hermes code 20305, Universidad Nacional de Colombia, Sede Medellín, Colombia.
¹ Institut für Mathematik, Berlin, Germany
e-mail: pommeren@math.tu-berlin.de
² Universidad Nacional de Colombia, Medellín, Colombia
e-mail: mmtoro@unal.edu.co

Abstract

We study free subgroups of index four of the parametrized modular group Π, the subgroup of SL generated by and . There are eight free subgroups, four of which are normal and four are non-normal. Then we study the intersections of the normal subgroups. We give canonical presentations in terms of generators and relations. At the end of the paper we study connections between Π and the Bianchi groups, the two-parabolic group and a group from relativity theory.

Key words and phrases. Parametrized modular group, free subgroups, Bianchi groups, Picard group, discrete relativity theory.

2010 Mathematics Subject Classification. 11R65, 14C22.

Resumen

Estudiamos los subgrupos libres de índice cuatro del grupo modular parametrizado Π, que es el subgrupo de SL generado por y . Hay ocho subgrupos libres, cuatro de los cuales son normales y los otros cuatro no lo son. Luego estudiamos las intersecciones de estos subgrupos. Damos presentaciones canónicas en término de generadores y relaciones. Al final del artículo estudiamos conexiones entre Π y los grupos de Bianchi, el grupo dos-parabólico y un grupo de la teoría de la relatividad.

Palabras y frases clave. Grupo modular parametrizado, subgrupos libres, grupos de Bianchi, grupo de Picard, teoría de la relatividad discreta.

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References

[1] G. H. Burde and H. Zieschang, Knots, Walter de Gruyter, 1985.         [ Links ]

[2] B. Fine, Algebraic theory of Bianchi groups, Marcel Dekker, New York, 1989.         [ Links ]

[3] B. Fine and M. Newman, The normal subgroup structure of the picard group, Trans. Amer. Math. Soc. 302 (1987), 769-786.         [ Links ]

[4] G. Jensen and Ch. Pommerenke, Discrete space-time and Lorentz transformations, to appear 2015 in the Canadian Mathematical Bulletin.         [ Links ]

[5] N. Yílmaz and N. Cangül, Conjugacy classes of elliptic elements in the Picard group, Turk. J. Math. 24 (2000), 209-220.         [ Links ]

[6] M. Lorente and P. Kramer, Representations of the discrete inhomogeneous Lorentz group and Dirac wave equation on the lattice, J. Phys A. 32 (1999), 2481-2497.         [ Links ]

[7] R. C. Lyndon and P. E. Schupp, Combinatorial group theory, Springer, New York, 1977.         [ Links ]

[8] C. MacLachlan and A. W. Reid, The arithmetic of hyperbolic 3-manifolds, Springer, New York, 2003.         [ Links ]

[9] W. Magnus, A. Karrass, and D. Solitar, Combinatorial group theory, 2. ed, Dover Publ., New York, 1976.         [ Links ]

[10] D. Mejia, Ch. Pommerenke, and M. Toro, On the parametrized modular group, to appear in J. Analyse Math.         [ Links ]

[11] Ch. Pommerenke and M. Toro, On the two-parabolic subgroups of SL2, C, Rev. Col. Mat. 45 (2011), 37-50.         [ Links ]

[12] R. Riley, Parabolic representations of knot groups I, Proc. London Math. Soc. 3 (1972), 217-242.         [ Links ]

[13] A. Schild, Discrete space-time and integral Lorentz transformations, Canad. J. Math. 1 (1949), 29-47.         [ Links ]

[14] R. G. Swan, Generators and relations for certain special linear groups, Advances Math. 6 (1971), 1-77.         [ Links ]

(Recibido en mayo de 2015. Aceptado en octubre de 2015)